Let's analyze the features of the function $f(x) = -2 \log_2 (x - 2)$.

1. Type of Function:

   • Since the function involves a logarithmic term, $f(x)$ is a logarithmic function.

2. Asymptote:

   • The argument of the logarithmic function, $x - 2$, must be positive for the function to be defined, which implies $x > 2$.
   • Therefore, the vertical asymptote occurs at $x = 2$.

3. Range:

   • The function $f(x) = -2 \log_2 (x - 2)$ can take all real values as $x$ approaches 2 (from the right) and extends to infinity. Since it has no upper or lower bounds in the $y$-direction, the range is $(-\infty, \infty)$.

4. Domain:

   • Since the logarithm is only defined for positive arguments, $x - 2 > 0 \Rightarrow x > 2$.
   • The domain of the function is $(2, \infty)$.

5. End Behavior:

   • Left Side (as $x \to 2^+$): $f(x) \to \infty$ because $\log_2(x - 2)$ approaches negative infinity, but the negative sign in front of the function makes it approach positive infinity.
   • Right Side (as $x \to \infty$): $f(x) \to -\infty$ as $x$ increases, making $\log_2(x - 2)$ approach infinity, and the negative sign causes $f(x)$ to approach negative infinity.

Given this information, here’s how to fill in the blanks in the answer template:

• The function $f(x)$ is a logarithmic function with a vertical asymptote of $x = 2$.
• The range of the function is $(-\infty, \infty)$, and it is decreasing on its domain of $(2, \infty)$.
• The end behavior on the LEFT side is as $x \to 2^+$, $f(x) \to \infty$.
• The end behavior on the RIGHT side is as $x \to \infty$, $f(x) \to -\infty$.

Would you like more details on any of these steps or have further questions?

Here are some related questions for further understanding:

1. How does the base of the logarithm affect the shape of the graph?
2. Why does the function approach positive infinity near the vertical asymptote?
3. What would happen if the coefficient of the logarithm were positive instead of negative?
4. How would changing the argument from $x - 2$ to $x + 2$ affect the domain and asymptote?
5. What is the significance of the vertical asymptote in logarithmic functions?

Tip: In logarithmic functions, the argument inside the logarithm determines the location of the vertical asymptote.